To estimate each candidate SIM, we follow [1]} to achieve the identification of \({\beta }_{(s)}\)  by normalizing its first element to 1 and employ the nonlinear least squares (NLS). One of the advantages of NLS is its light computation burden, which is crucial in our case since our averaging technique is based on a cross-validation criterion (discussed in detail below) and the number of candidate models is typically substantial.
To define the NLS estimator, let \(k(\cdot )\)  be a kernel function. For the \(s^{\mbox{\tiny \rm th}}\)  candidate model, denote \(h_s\)  as the bandwidth, \(k_{h_s}(\cdot )=k(\cdot /h_s)/h_s\) , and \({\bf K}_{(s)}({\beta }_{(s)})=\lbrace K_{{(s)},ij}({\beta }_{(s)})\rbrace _{n\times n}\)  as an \(n\times n\)  smoother matrix with the typical element
\(K_{{(s)},ij}({\beta }_{(s)})=k_{h_s}({\bf x}_{{(s)},i}^{\rm T}{\beta }_{(s)}-{\bf x}_{{(s)},j}^{\rm T}{\beta }_{(s)})/\sum \nolimits _{j^\ast =1}^nk_{h_s}({\bf x}_{{(s)},i}^{\rm T}{\beta }_{(s)}-{\bf x}_{{(s)},j^\ast }^{\rm T}{\beta }_{(s)}).\nonumber \) 
